While there are several proposal for alleviating sign problems proposed, one long-standing method which successfully tamed sign problems for some models of our interest is the so-called manifold deformation method [1]}. The idea of the method is simple: we deform the contour of integration is the path integral, \(\mathbb {R}^N\)  to the complex plane \(\mathbb {C}^N\)  of the field variables \(\phi \) , aiming for a larger average sign, and thus milder sign problem. When the average sign is exactly 1, the sign problem is solved — the average sign stays exactly 1 no matter how large the size of the lattice becomes.
